In this paper we analyse a pressure stabilized, finite element method for the
unsteady, incompressible Navier-Stokes equations in primitive variables; for the
time discretization we focus on a fully implicit, monolithic scheme. We provide
some error estimates for the fully discrete solution which show that the velocity
is first order accurate in the time step and attains optimal order accuracy in the
mesh size for the given spatial interpolation, both in the spaces L^2(W) and
H(W) the pressure solution is shown to be order 1/2 accurate in the time step and also
optimal in the mesh size. These estimates are proved assuming only a weak
compatibility condition on the approximating spaces of velocity and pressure,
which is satisfied by equal order interpolations.